In conventional machines, the guidance device generally comprises a fixed part with two vee-shaped flat sides in which the component to be machined is engaged and guided and a mobile part, the support finger, which presses the component into the vee while ensuring rotation, the quality of which only depends at that point on the circularity of the component to be machined itself. During a change of series, the positioning vee must be changed, since the tool diameter of the new series differs from that of the preceding series and its positioning height must be adjusted with great precision in order to guarantee a perfect position of the rotation axis. This adjustment work for the position of the vee, according to a minimum of three degrees of freedom, is time consuming, thereby reducing the productivity of the whole machine. Finally, it reduces the versatility of the production system by only making it attractive for very long series.